1 | /* log2l.c |
2 | * Base 2 logarithm, 128-bit long double precision |
3 | * |
4 | * |
5 | * |
6 | * SYNOPSIS: |
7 | * |
8 | * long double x, y, log2l(); |
9 | * |
10 | * y = log2l( x ); |
11 | * |
12 | * |
13 | * |
14 | * DESCRIPTION: |
15 | * |
16 | * Returns the base 2 logarithm of x. |
17 | * |
18 | * The argument is separated into its exponent and fractional |
19 | * parts. If the exponent is between -1 and +1, the (natural) |
20 | * logarithm of the fraction is approximated by |
21 | * |
22 | * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x). |
23 | * |
24 | * Otherwise, setting z = 2(x-1)/x+1), |
25 | * |
26 | * log(x) = z + z^3 P(z)/Q(z). |
27 | * |
28 | * |
29 | * |
30 | * ACCURACY: |
31 | * |
32 | * Relative error: |
33 | * arithmetic domain # trials peak rms |
34 | * IEEE 0.5, 2.0 100,000 2.6e-34 4.9e-35 |
35 | * IEEE exp(+-10000) 100,000 9.6e-35 4.0e-35 |
36 | * |
37 | * In the tests over the interval exp(+-10000), the logarithms |
38 | * of the random arguments were uniformly distributed over |
39 | * [-10000, +10000]. |
40 | * |
41 | */ |
42 | |
43 | /* |
44 | Cephes Math Library Release 2.2: January, 1991 |
45 | Copyright 1984, 1991 by Stephen L. Moshier |
46 | Adapted for glibc November, 2001 |
47 | |
48 | This library is free software; you can redistribute it and/or |
49 | modify it under the terms of the GNU Lesser General Public |
50 | License as published by the Free Software Foundation; either |
51 | version 2.1 of the License, or (at your option) any later version. |
52 | |
53 | This library is distributed in the hope that it will be useful, |
54 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
55 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
56 | Lesser General Public License for more details. |
57 | |
58 | You should have received a copy of the GNU Lesser General Public |
59 | License along with this library; if not, see <https://www.gnu.org/licenses/>. |
60 | */ |
61 | |
62 | #include <math.h> |
63 | #include <math_private.h> |
64 | #include <libm-alias-finite.h> |
65 | |
66 | /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x) |
67 | * 1/sqrt(2) <= x < sqrt(2) |
68 | * Theoretical peak relative error = 5.3e-37, |
69 | * relative peak error spread = 2.3e-14 |
70 | */ |
71 | static const _Float128 P[13] = |
72 | { |
73 | L(1.313572404063446165910279910527789794488E4), |
74 | L(7.771154681358524243729929227226708890930E4), |
75 | L(2.014652742082537582487669938141683759923E5), |
76 | L(3.007007295140399532324943111654767187848E5), |
77 | L(2.854829159639697837788887080758954924001E5), |
78 | L(1.797628303815655343403735250238293741397E5), |
79 | L(7.594356839258970405033155585486712125861E4), |
80 | L(2.128857716871515081352991964243375186031E4), |
81 | L(3.824952356185897735160588078446136783779E3), |
82 | L(4.114517881637811823002128927449878962058E2), |
83 | L(2.321125933898420063925789532045674660756E1), |
84 | L(4.998469661968096229986658302195402690910E-1), |
85 | L(1.538612243596254322971797716843006400388E-6) |
86 | }; |
87 | static const _Float128 Q[12] = |
88 | { |
89 | L(3.940717212190338497730839731583397586124E4), |
90 | L(2.626900195321832660448791748036714883242E5), |
91 | L(7.777690340007566932935753241556479363645E5), |
92 | L(1.347518538384329112529391120390701166528E6), |
93 | L(1.514882452993549494932585972882995548426E6), |
94 | L(1.158019977462989115839826904108208787040E6), |
95 | L(6.132189329546557743179177159925690841200E5), |
96 | L(2.248234257620569139969141618556349415120E5), |
97 | L(5.605842085972455027590989944010492125825E4), |
98 | L(9.147150349299596453976674231612674085381E3), |
99 | L(9.104928120962988414618126155557301584078E2), |
100 | L(4.839208193348159620282142911143429644326E1) |
101 | /* 1.000000000000000000000000000000000000000E0L, */ |
102 | }; |
103 | |
104 | /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), |
105 | * where z = 2(x-1)/(x+1) |
106 | * 1/sqrt(2) <= x < sqrt(2) |
107 | * Theoretical peak relative error = 1.1e-35, |
108 | * relative peak error spread 1.1e-9 |
109 | */ |
110 | static const _Float128 R[6] = |
111 | { |
112 | L(1.418134209872192732479751274970992665513E5), |
113 | L(-8.977257995689735303686582344659576526998E4), |
114 | L(2.048819892795278657810231591630928516206E4), |
115 | L(-2.024301798136027039250415126250455056397E3), |
116 | L(8.057002716646055371965756206836056074715E1), |
117 | L(-8.828896441624934385266096344596648080902E-1) |
118 | }; |
119 | static const _Float128 S[6] = |
120 | { |
121 | L(1.701761051846631278975701529965589676574E6), |
122 | L(-1.332535117259762928288745111081235577029E6), |
123 | L(4.001557694070773974936904547424676279307E5), |
124 | L(-5.748542087379434595104154610899551484314E4), |
125 | L(3.998526750980007367835804959888064681098E3), |
126 | L(-1.186359407982897997337150403816839480438E2) |
127 | /* 1.000000000000000000000000000000000000000E0L, */ |
128 | }; |
129 | |
130 | static const _Float128 |
131 | /* log2(e) - 1 */ |
132 | LOG2EA = L(4.4269504088896340735992468100189213742664595E-1), |
133 | /* sqrt(2)/2 */ |
134 | SQRTH = L(7.071067811865475244008443621048490392848359E-1); |
135 | |
136 | |
137 | /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ |
138 | |
139 | static _Float128 |
140 | neval (_Float128 x, const _Float128 *p, int n) |
141 | { |
142 | _Float128 y; |
143 | |
144 | p += n; |
145 | y = *p--; |
146 | do |
147 | { |
148 | y = y * x + *p--; |
149 | } |
150 | while (--n > 0); |
151 | return y; |
152 | } |
153 | |
154 | |
155 | /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ |
156 | |
157 | static _Float128 |
158 | deval (_Float128 x, const _Float128 *p, int n) |
159 | { |
160 | _Float128 y; |
161 | |
162 | p += n; |
163 | y = x + *p--; |
164 | do |
165 | { |
166 | y = y * x + *p--; |
167 | } |
168 | while (--n > 0); |
169 | return y; |
170 | } |
171 | |
172 | |
173 | |
174 | _Float128 |
175 | __ieee754_log2l (_Float128 x) |
176 | { |
177 | _Float128 z; |
178 | _Float128 y; |
179 | int e; |
180 | int64_t hx, lx; |
181 | |
182 | /* Test for domain */ |
183 | GET_LDOUBLE_WORDS64 (hx, lx, x); |
184 | if (((hx & 0x7fffffffffffffffLL) | lx) == 0) |
185 | return (-1 / fabsl (x)); /* log2l(+-0)=-inf */ |
186 | if (hx < 0) |
187 | return (x - x) / (x - x); |
188 | if (hx >= 0x7fff000000000000LL) |
189 | return (x + x); |
190 | |
191 | if (x == 1) |
192 | return 0; |
193 | |
194 | /* separate mantissa from exponent */ |
195 | |
196 | /* Note, frexp is used so that denormal numbers |
197 | * will be handled properly. |
198 | */ |
199 | x = __frexpl (x, &e); |
200 | |
201 | |
202 | /* logarithm using log(x) = z + z**3 P(z)/Q(z), |
203 | * where z = 2(x-1)/x+1) |
204 | */ |
205 | if ((e > 2) || (e < -2)) |
206 | { |
207 | if (x < SQRTH) |
208 | { /* 2( 2x-1 )/( 2x+1 ) */ |
209 | e -= 1; |
210 | z = x - L(0.5); |
211 | y = L(0.5) * z + L(0.5); |
212 | } |
213 | else |
214 | { /* 2 (x-1)/(x+1) */ |
215 | z = x - L(0.5); |
216 | z -= L(0.5); |
217 | y = L(0.5) * x + L(0.5); |
218 | } |
219 | x = z / y; |
220 | z = x * x; |
221 | y = x * (z * neval (z, R, 5) / deval (z, S, 5)); |
222 | goto done; |
223 | } |
224 | |
225 | |
226 | /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ |
227 | |
228 | if (x < SQRTH) |
229 | { |
230 | e -= 1; |
231 | x = 2.0 * x - 1; /* 2x - 1 */ |
232 | } |
233 | else |
234 | { |
235 | x = x - 1; |
236 | } |
237 | z = x * x; |
238 | y = x * (z * neval (x, P, 12) / deval (x, Q, 11)); |
239 | y = y - 0.5 * z; |
240 | |
241 | done: |
242 | |
243 | /* Multiply log of fraction by log2(e) |
244 | * and base 2 exponent by 1 |
245 | */ |
246 | z = y * LOG2EA; |
247 | z += x * LOG2EA; |
248 | z += y; |
249 | z += x; |
250 | z += e; |
251 | return (z); |
252 | } |
253 | libm_alias_finite (__ieee754_log2l, __log2l) |
254 | |